Modify ACVaR to satisfy all coherence axioms

Construct a modified ACVaR definition for finite-state Markov chains that satisfies all four coherence axioms of Artzner, Delbaen, Eber, and Heath (translation invariance, positive homogeneity, monotonicity, and subadditivity), thereby ensuring convexity.

Background

The proposed ACVaR satisfies several coherence properties (translation invariance, positive homogeneity, and monotonicity) but may fail subadditivity and thus convexity in general. Achieving full coherence would strengthen its theoretical grounding as a risk measure.

The authors explicitly raise whether the definition can be adjusted to satisfy all the axioms of coherent risk measures per Artzner et al. (1999). This entails identifying modifications that restore subadditivity without losing the asymptotic and large-deviations-based interpretation.